Math Glossary - Mathematics Definitions Math Glossary - Mathematics Definitions Math Glossary - Mathematics Definitions Look Up the Meaning of Math Words Math is already its own language, so make sure you know the meaning of words used to describe it!. RunPhoto, Getty Images By Anne Helmenstine This is a glossary of common math terms used in arithmetic, geometry, algebra, and statistics. Abacus - An early counting tool used for basic arithmetic. Absolute Value - Always a positive number, refers to the distance of a number from 0, the distances are positive. Acute Angle - The measure of an angle with a measure between 0° and 90° or with less than 90° radians. Addend - A number which is involved in addition. Numbers being added are considered to be the addends. Bisector Box and Whisker Plot/Chart - A graphical representation of data that spots differences in distributions. Plots the ranges of data sets. Calculus - The branch of mathematics involving derivatives and integrals. The study of motion in which changing values are studied. continue reading below our video Common Mathematical Symbols Capacity - The amount a container will hold. Centimeter - A measure of length. 2.5cm is approximately an inch.

A metric unit of measurement. Circumference - The complete distance around a circle or a square. Chord - The segment which joins two points on a circle. Coefficient - A factor of the term. x is the coefficient in the term x(a + b) or 3 is the coefficient in the term 3y. Common Factors - A factor of two or more numbers. A number that will divide exactly into different numbers. Complementary Angles - The two angles involved when the sum is 90°. Composite Number - A composite number has at least one other factor aside from its own. A composite number cannot be a prime number. Cone - A three dimensional shape with only one vertex, having a circular base. Conic Section - The section formed by the intersection of a plane and a cone. Constant - A value that doesn't change. Coordinate - The ordered pair that states the location on a coordinate plane. Used to describe location and or position. Congruent - Objects and figures that have the same size and shape. The shapes can be turned into one another with a flip, rotation or turn.

Cosine - The ratio of the length (in a right triangle) of the side adjacent to an acute angle to the length of the hypotenuse Cylinder - A Three dimensional shape with a parallel circle and each end and joined by a curved surface. Decagon - A polygon/shape that has ten angles and ten straight lines. Decimal - A real number on the base ten standard numbering system. Denominator - The denominator is the bottom number of a fraction. (Numerator is the top number) The Denominator is the total number of parts. Degree - The unit of an angle, angles are measured in degrees shown by the degree symbol: ° Diagonal - A line segment that connects two vertices in a polygon. Diameter - A chord that passes through the center of a circle. Also the length of a line that cuts the shape in half. Difference - The difference is what is found when one number is subtracted from another. Finding the difference in a number requires the use of subtraction. Digit - Digits are making reference to numerals. 176 is a 3 digit number. Dividend - The number that is being divided. The number found inside the bracket. Divisor - The number that is doing the dividing.

The number found outside of the division bracket. Edge - A line that joins a polygon or the line (edge) where two faces meet in a 3 dimensional solid. Ellipse - An ellipse looks like a slightly flattened circle. A plane curve. Orbits take the form of ellipses. End Point - The 'point' at which a line or a curve ends. Equilateral - All sides are equal. Equation - A statement showing the equality of two expressions usually separated by left and right signs and joined by an equals sign. Even Number - A number that can be divided or is divisible by 2. Event - Often refers to the outcome of probability. Answers questions like 'What is the probability the spinner will land on red?' Evaluate - To calculate the numerical value. Exponent - The number that gives reference to the repeated multiplication required. The exponent of 34 is the 4. Expressions - Symbols that represent numbers or operations. A way of writing something that uses numbers and symbols. Face - The face refers to the shape that is bounded by the edges on a 3 dimensional object. Factor - A number that will divide into another number exactly. (The factors of 10 are 1, 2 and 5).

Factoring - The process of breaking numbers down into all of their factors. Factorial Notation - Often in combinatorics, you will be required to multiply consecutive numbers. The symbol used in factorial notation is ! When you see x!, the factorial of x is needed. Factor Tree - A graphical representation showing the factors of a specific number. Fibonacci Sequence - A sequence whereby each number is the sum of the two numbers preceding it. Figure - Two dimensional shapes are often referred to as figures. Finite - Not infinite. Finite has an end. Flip - A reflection of a two dimensional shape, a mirror image of a shape. Formula - A rule that describes the relationship of two or more variables. An equation stating the rule. Fraction - A way of writing numbers that are not whole numbers. The fraction is written like 1/2. Frequency - The number of times an event can happen in a specific period of times. Often used in probability. Furlong - A unit of measurement - the side length of one square of an acre. One furlong is approximately 1/8 of a mile, 201.17 meters and 220 yards. Geometry - The study of lines, angles, shapes and their properties.

Geometry is concerned with physical shapes and the dimensions of the objects. Graphing Calculator - A larger screen calculator that's capable of showing/drawing graphs and functions. Graph Theory - A branch of mathematics focusing on the properties of a variety of graphs. Greatest Common Factor - The largest number common to each set of factors that divides both numbers exactly. E.g., the greatest common factor of 10 and 20 is 10. Hexagon - A six sided and six angled polygon. Hex means 6. Histogram - A graph that uses bars where each bar equals a range of values. Hyperbola - One type of conic section. The hyperbola is the set of all points in a plane. The difference of whose distance from two fixed points in the plane is the positive constant. Hypotenuse - The longest side of a right angled triangle. Always the side that's opposite of the right angle. Identity - An equation that is true for values of their variables. Improper Fraction - A fraction whereby the denominator is equal to or greater than the numerator. E.g., 6/4 Inequality - A mathematical equation containing either a greater than, less than or not equal to symbols.

Integers - Whole numbers, positive or negative including zero. Irrational - A number that cannot be represented as a decimal or as a fraction. A number like pi is irrational because it contains an infinite number of digits that keep repeating, many square roots are irrational numbers. Isoceles - A polygon having two sides equal in length. Kilometer - A unit of measure that equals 1000 meters. Knot - A curve formed by an interlacing piece of spring by joining the ends. Like Terms - Terms with the same variable and the same exponents/degrees. Like Fractions - Fractions having the same denominator. (Numerator is the top, denominator is the bottom) Line - A straight infinite path joining an infinite number of points. The path can be infinite in both directions. Line Segment - A straight path that has a beginning and an end - endpoints. Linear Equation - An equation whereby letters represent real numbers and whose graph is a line. Line of Symmetry - A line that divides a figure or shape into two parts. The two shape must equal one another. Logic - Sound reasoning and the formal laws of reasoning.

Logarithm - A power to which a base, [actually 10] must be raised to produce a given number. If nx = a, the logarithm of a, with n as the base, is x. Mean - The mean is the same as the average. Add up the series of numbers and divide the sum by the number of values. Median - The Median is the 'middle value' in your list or series of numbers. When the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order. When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two. Midpoint - A point that is exactly half way between two set points. Mixed Numbers - Mixed numbers refer to whole numbers with fractions or decimals. Example 3 1/2 or 3.5. Mode - The mode in a list of numbers refers to the list of numbers that occur most frequently. A trick to remember this one is to remember that mode starts with the same first two letters that most does. Most frequently - Mode.

Modular Arithmetic - a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value of the modulus. Monomial - An algebraic expression consisting of a single term. Multiple - The multiple of a number is the product of the number and any other whole number. (2,4,6,8 are multiples of 2) Multiplication - Often referred to as 'fast adding'. Multiplication is the repeated addition of the same number 4x3 is the same as saying 3+3+3+3. Multiplicand - A quantity multiplied by another. A product is obtained by multiplying two or more multiplicands. Natural Numbers - Regular counting numbers. Negative Number - A number less than zero. For instance - a decimal .10 Net - Often referred to in elementary school math. A flattened 3-D shape that can be turned into a 3-D object with glue/tape and folding. Nth Root - The nth root of a number is the number needed to multiply by itself 'n' times in order to get that number. For instance: the 4th root of 3 is 81 because 3 X 3 X 3 X 3 = 81.

Norm - The mean or the average - an established pattern or form. Numerator - The top number in a fraction. In 1/2, 1 is the numerator and 2 is the denominator. The numerator is the portion of the denominator. Number Line - A line in which points all correspond to numbers. Numeral - A written symbol referring to a number. Obtuse Angle - An angle having a measure greater than 90° and up to 180°. Obtuse Triangle - A triangle with at least one obtuse angle as described above. Octagon - A polygon with 8 sides. Odds - The ratio/likelihood of an event in probability happening. The odds of flipping a coin and having it land on heads has a 1-2 chance. Odd Number - A whole number that is not divisible by 2. Operation - Refers to either addition, subtraction, multiplication or division which are called the four operations in mathematics or arithmetic. Ordinal - Ordinal numbers refer to the position: first, second, third etc. Order of Operations - A set of rules used to solve mathematical problems. BEDMAS is often the acronym used to remember the order of operations.

BEDMAS stands for 'brackets, exponents, division, multiplication, addition and subtraction. Outcome - Used usually in probability to refer to the outcome of an event. Parallelogram - A quadrilateral that has both sets of opposite sides that are parallel. Parabola - A type of curve, any point of which is equally distant from a fixed point, called the focus, and a fixed straight line, called the directrix. Pentagon - A five sided polygon. Regular pentagons have five equal sides and five equal angles. Percent - A ratio or fraction in which the second term on denominator is always 100. Perimeter - The total distance around the outside of a polygon. The total distance around is obtained by adding together the units of measure from each side. Perpendicular - When two lines or line segments intersect and form right angles. Pi p - The symbol for Pi is actually a greek letter. Pi is used to represent the ratio of a circumference of a circle to its diameter. Plane - When a set of points joined together form a flat surface, the plan can extend without end in all directions. Polynomial - An algebraic term. The sum of 2 or more monomials.

Polynomials include variables and always have one or more terms. Polygon - Line segments joined together to form a closed figure. Rectangles, squares, pentagons are all examples of polygons. Prime Numbers - Prime numbers are integers that are greater than 1 and are only divisible by themselves and 1. Probability - The likelihood of an event happening. Product - The sum obtained when any two or more numbers are multiplied together. Proper Fraction - A fraction where the denominator is greater than the numerator. Protractor - A semi-circle device used for measuring angles. The edge is subdivided into degrees. Quadrant - One quarter (qua) of the plane on the Cartesian coordinate system. The plane is divided into 4 sections, each section is called a quadrant. Quadratic Equation - An equation that can be written with one side equal to 0. Asks you to find the quadratic polynomial that is equal to zero. Quadrilateral - A four (quad) sided polygon/shape. Quadruple - To multiply or to be multiplied by 4. Qualitative - A general description of properties that cannot be written in numbers. Quartic - A polynomial having a degree of 4. Quintic - A polynomial having a degree of 5.

Quotient - The solution to a division problem. Radius - A line segment from the center of a circle to any point on the circle. Or the line from the center of a sphere to any point on the outside edge of the sphere. The radius is the distance from the center of a circle/sphere to the outside edge. Ratio - The relation between to quantities. Ratios can be expressed in words, fractions, decimals or percents. E.g., the ratio given when a team wins 4 out of 6 games can be said a 4:6 or four out of six or 4/6. Ray - A straight line with one endpoint. The line extends infinitely. Range - The difference between the maximum and the minimum in a set of data. Rectangle - A parallelogram which has four right angles. Repeating Decimal - A decimal with endlessly repeating digits. E.g., 88 divided by 33 will give a 2.6666666666666 Reflection - A mirror image of a shape or an object. Obtained from flipping the image/object. Remainder - The number that is left over when the number cannot be divided evenly into the number.

Right Angle - An angle that is 90°. Right Triangle - A triangle having one angle equal to 90°. Rhombus - A parallelogram with four equal sides, sides are all the same length. Scalene Triangle - A triangle with 3 unequal sides. Secant Sector - An area between an arc and two radii of a circle. Sometimes referred to as a wedge. Slope - The slope shows the steepness or incline of a line, determined from two points on the line. Square Root- To square a number, you multiply it by itself. The square root of a number is the value of the number when multiplied by itself, gives you the original number. For instance 12 squared is 144, the square root of 144 is 12. Stem and Leaf - A graphic organizer to organize and compare data. Similar to a histogram, organizes intervals or groups of data. Subtraction - The operation of finding the difference between two numbers or quantities. A process of 'taking away'. Supplementary Angles - Two angles are supplementary if their sum totals 180°. Symmetry - Two halves which match perfectly.

Tangent - When an angle in a right angle is X, the tangent of x is the ratio of lengths of the side opposite x to the side adjacent to x. Term - A part of an algebraic equation or a number in a sequence or a series or a product of real numbers and/or variables. Tessellation - Congruent plane figures/shapes that cover a plane completely without overlapping. Translation - A term used in geometry. Often called a slide. The figure or shape is moved from each point of the figure/shape in the same direction and distance. Transversal - A line that crosses/intersects two or more lines. Trapezoid - A quadrilateral with exactly two parallel sides. Tree Diagram - Used in probability to show all of the possible outcomes or combinations of an event. Triangle - Three sided polygon. Trinomial - An algebraic equation with 3 terms - polynomial . Unit - A standard quantity used in measurement. An inch is a unit of length, a centimeter is a unit of length a pound is a unit of weight. Uniform - All the same. Having the same in size, texture, color, design etc. Variable - When a letter is used to represent a number or number in equations and or expressions.

E.g., in 3x + y, both y and x are the variables. Venn Diagram - A Venn diagram is often two circles (can be other shapes) that overlap. The overlapping part usually contains information that is pertinent to the labels on both sides of the Venn diagram. For instance: one circle could be labeled 'Odd Numbers', the other circle could be labeled 'Two Digit Numbers' the overlapping portion must contain numbers that are odd and have two digits. Thus, the overlapping portions shows the relationship between the sets. (Can be more than 2 circles.) Volume - A unit of measure. The amount of cubic units that occupy a space. A measurement of capacity or volume. Vertex- A point of intersection where two (or more) rays meet, often called the corner. Wherever sides or edges meet on polygons or shapes. The point of a cone, the corners of cubes or squares. Weight - A measure of how heavy something is. Whole Number - A whole number doesn't contain a fraction. A whole number is a positive integer which has 1 or more units and can be positive or negative. X-Axis - The horizontal axis in a coordinate plane.

X-Intercept - The value of X when the line or curve intersects or crosses the x axis. X - The roman numeral for 10. x - A symbol most often used to represent an unknown quantity in an equation. Y-Axis - The vertical axis in a coordinate plane. Y-Intercept - The value of y when the line or curve intersects or crosses the y axis. Yard- A unit of measure. A yard is approximately 91.5 cm. A yard is also 3 feet. Mean, Median and Mode Year 8 Interactive Maths - Second Edition Mean, Median and Mode We use statistics such as the mean , median and mode to obtain information about a population from our sample set of observed values. Mean The mean (or average) of a set of data values is the sum of all of the data values divided by the number of data values. That is: Example 1 The marks of seven students in a mathematics test with a maximum possible mark of 20 are given below: 15 13 18 16 14 17 12 Find the mean of this set of data values. Solution: So, the mean mark is 15.

Symbolically, we can set out the solution as follows: So, the mean mark is 15. Median The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value. Example 2 The marks of nine students in a geography test that had a maximum possible mark of 50 are given below: 47 35 37 32 38 39 36 34 35 Find the median of this set of data values. Solution: Arrange the data values in order from the lowest value to the highest value: 32 34 35 35 36 37 38 39 47 The fifth data value, 36, is the middle value in this arrangement. Note: In general: If the number of values in the data set is even, then the median is the average of the two middle values.

Example 3 Find the median of the following data set: 12 18 16 21 10 13 17 19 Solution: Arrange the data values in order from the lowest value to the highest value: 10 12 13 16 17 18 19 21 The number of values in the data set is 8, which is even. So, the median is the average of the two middle values. Alternative way: There are 8 values in the data set. The fourth and fifth scores, 16 and 17, are in the middle. That is, there is no one middle value. Note: Half of the values in the data set lie below the median and half lie above the median. The median is the most commonly quoted figure used to measure property prices. The use of the median avoids the problem of the mean property price which is affected by a few expensive properties that are not representative of the general property market. Mode The mode of a set of data values is the value(s) that occurs most often. The mode has applications in printing.

For example, it is important to print more of the most popular books; because printing different books in equal numbers would cause a shortage of some books and an oversupply of others. Likewise, the mode has applications in manufacturing. For example, it is important to manufacture more of the most popular shoes; because manufacturing different shoes in equal numbers would cause a shortage of some shoes and an oversupply of others. Example 4 Find the mode of the following data set: 48 44 48 45 42 49 48 Solution: The mode is 48 since it occurs most often. Note: It is possible for a set of data values to have more than one mode. If there are two data values that occur most frequently, we say that the set of data values is bimodal. If there is no data value or data values that occur most frequently, we say that the set of data values has no mode. Analysing Data The mean , median and mode of a data set are collectively known as measures of central tendency as these three measures focus on where the data is centred or clustered.

To analyse data using the mean, median and mode, we need to use the most appropriate measure of central tendency. The following points should be remembered: The mean is useful for predicting future results when there are no extreme values in the data set. However, the impact of extreme values on the mean may be important and should be considered. E.g. The impact of a stock market crash on average investment returns. The median may be more useful than the mean when there are extreme values in the data set as it is not affected by the extreme values. The mode is useful when the most common item, characteristic or value of a data set is required. Key Terms Statistics Year 10 Interactive Maths - Second Edition Statistics A population is the whole set of items from which a data sample can be drawn; so a sample is only a portion of the population data. Features of a sample are described by statistics. Statistical methods enable us to arrange, analyse and interpret the sample data obtained from a population. We gather sample data when it is impractical to analyse the population data as a smaller sample often allows us to gain a better understanding of the population without doing too much work or wasting precious time.

For example, television program ratings indicate the number of TV viewers per 100 watching various programs. Ratings are determined from a small sample of viewers that represent the entire audience (or population) as it is too expensive to monitor every household that watches television. The small sample gives a reasonable estimate of the entire population. Statistics are used for many purposes including election polling, sample marketing, understanding internet website visitors etc. In this chapter, we will consider summary statistics , tables , boxplots , scatterplots , line of best fit and time series data. Summary Statistics A set of data values collected by a survey or experiment is known as a sample. Summary statistics provide useful information about sample data. The most commonly used types of summary statistics are measures of location (or central tendency) and measures of spread . Measures of Location (or Central Tendency) We can describe a data sample's location by using statistics to analyse the sample's average, middle or most common value. The statistics used are known as the mean , median and mode and are collectively known as measures of location (or central tendency). Mean The mean of a set of values is defined as the sum of all the values divided by the number of values.

That is: Example 1 The marks of five candidates in a mathematics test with a maximum possible mark of 20 are given below. 15 13 19 18 14 Find the mean value. The marks awarded to seven pupils for an assignment were as follows: 19 15 19 16 13 20 19 a. Find the median mark. b. State the mode. a. Arrange the marks in ascending order of magnitude: 13 15 16 19 19 19 20 Note: The fourth score, 19, is the middle data value in this arrangement. b. 19 is the score that occurs most often. Note the following: The mode has applications in manufacturing. For example, it is important to manufacture more of the most popular cars, because manufacturing different cars in equal numbers would cause a shortage of some cars and an oversupply of others. The mean (or average) of a set of data values is used to predict future results. The median is easy to comprehend. Extreme values in the data set do not affect the median. Key Terms Statistics: Power from Data!

Range and quartiles Semi-quartile range Range The range is very easy to calculate because it is simply the difference between the largest and the smallest observed values in a data set. Thus, range, including any outliers, is the actual spread of data. Range = difference between highest and lowest observed values A great deal of information is ignored when computing the range, since only the largest and smallest data values are considered. The range value of a data set is greatly influenced by the presence of just one unusually large or small value (outlier). The range can be expressed as an interval such as 4–10, where 4 is the lowest value and 10 is highest. Often, it is expressed as interval width. For example, the range of 4–10 can also be expressed as a range of 6. The latter convention will be used throughout this chapter. The disadvantage of using range is that it does not measure the spread of the majority of values in a data set—it only measures the spread between highest and lowest values. As a result, other measures are required in order to give a better picture of the data spread.

The range is an informative tool used as a supplement to other measures such as the standard deviation or semi-interquartile range, but it should rarely be used as the only measure of spread. Quartiles The median divides the data into two equal sets. For more information on the median, refer to the chapter on Measures of central tendency : The lower quartile is the value of the middle of the first set, where 25% of the values are smaller than Q1 and 75% are larger. This first quartile takes the notation Q1. The upper quartile is the value of the middle of the second set, where 75% of the values are smaller than Q3 and 25% are larger. This third quartile takes the notation Q3. It should be noted that the median takes the notation Q2, the second quartile.

Example 1 – Upper and lower quartiles Data: 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36 Ordered data: 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49 Median: 41 Interquartile range The interquartile range is another range used as a measure of the spread. The difference between upper and lower quartiles (Q3–Q1), which is called the interquartile range, also indicates the dispersion of a data set. The interquartile range spans 50% of a data set, and eliminates the influence of outliers because, in effect, the highest and lowest quarters are removed. Interquartile range = difference between upper quartile (Q3) and lower quartile (Q1) A year ago, Angela began working at a computer store. Her supervisor asked her to keep a record of the number of sales she made each month.

The following data set is a list of her sales for the last 12 months: 34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37. Use Angela's sales records to find: the median These results can be summarized as follows: Note: This example has an even number of observations. The median, Q2, lies between the centre of two observations (24 and 28), so the calculation of Q1 includes the observation 24 as it is below the value of Q2. Similarly, 28 is also included in the calculation of Q3 as it is above the value of Q2. Consider an odd number of observations such as 1, 2, 3, 4, 5, 6, 7. Here the value of Q2 is 4. As the location of the median is right on the fourth observation, this value is not included in calculating Q1 and Q3 , as we are interested only in the data above and below Q2.

In the above example, Q1 = 2 and Q3 = 6. Mean, Median, Mode, and Range | Purplemath Mean, Median, Mode, and Range Purplemath Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all. The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers. The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order from smallest to largest, so you may have to rewrite your list before you can find the median. The "mode" is the value that occurs most often. If no number in the list is repeated, then there is no mode for the list. Content Continues Below The "range" of a list a numbers is just the difference between the largest and smallest values.

Find the mean, median, mode, and range for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13 The mean is the usual average, so I'll add and then divide: (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15 Affiliate Note that the mean, in this case, isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.

The median is the middle value, so first I'll have to rewrite the list in numerical order: 13, 13, 13, 13, 14, 14, 16, 18, 21 There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number: 13, 13, 13, 13, 14, 14, 16, 18, 21 13, 13, 13, 13, 14, 14, 16, 18, 21 So the median is 14. The mode is the number that is repeated more often than any other, so 13 is the mode. The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.

mean: 15 mode: 13 range: 8 Note: The formula for the place to find the median is "([the number of data points] + 1) ÷ 2", but you don't have to use this formula. You can just count in from both ends of the list until you meet in the middle, if you prefer, especially if your list is short. Either way will work. Find the mean, median, mode, and range for the following list of values: 1, 2, 4, 7 mode: none range: 6 The values in the list above were all whole numbers, but the mean of the list was a decimal value. Getting a decimal value for the mean (or for the median, if you have an even number of data points) is perfectly okay; don't round your answers to try to match the format of the other numbers.

Content Continues Below Find the mean, median, mode, and range for the following list of values: 8, 9, 10, 10, 10, 11, 11, 11, 12, 13 The mean is the usual average, so I'll add up and then divide: (8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13) ÷ 10 = 105 ÷ 10 = 10.5 The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5-th value; the formula is reminding me, with that "point-five", that I'll need to average the fifth and sixth numbers to find the median. The fifth and sixth numbers are the last 10 and the first 11, so: (10 + 11) ÷ 2 = 21 ÷ 2 = 10.5 The mode is the number repeated most often.

This list has two values that are repeated three times; namely, 10 and 11, each repeated three times. The largest value is 13 and the smallest is 8, so the range is 13 – 8 = 5. mean: 10.5 modes: 10 and 11 range: 5 As you can see, it is possible for two of the averages (the mean and the median, in this case) to have the same value. But this is not usual, and you should not expect it. Advertisement WyzAnt Tutoring Note: Depending on your text or your instructor, the above data set may be viewed as having no mode rather than having two modes, because no single solitary number was repeated more often than any other. I've seen books that go either way on this; there doesn't seem to be a consensus on the "right" definition of "mode" in the above case. So if you're not certain how you should answer the "mode" part of the above example, ask your instructor before the next test. About the only hard part of finding the mean, median, and mode is keeping straight which "average" is which.

Just remember the following: mean: regular meaning of "average" median: middle value mode: most often (In the above, I've used the term "average" rather casually. The technical definition of what we commonly refer to as the "average" is technically called "the arithmetic mean": adding up the values and then dividing by the number of values. Since you're probably more familiar with the concept of "average" than with "measure of central tendency", I used the more comfortable term.) A student has gotten the following grades on his tests: 87, 95, 76, and 88. He wants an 85 or better overall. What is the minimum grade he must get on the last test in order to achieve that average? The minimum grade is what I need to find. To find the average of all his grades (the known ones, plus the unknown one), I have to add up all the grades, and then divide by the number of grades. Since I don't have a score for the last test yet, I'll use a variable to stand for this unknown value: "x".

Then computation to find the desired average is: (87 + 95 + 76 + 88 + x) ÷ 5 = 85 Multiplying through by 5 and simplifying, I get: 87 + 95 + 76 + 88 + x = 425 346 + x = 425 Hawaii Content & Performance Standards :: Standards Toolkit » Mathematics Grade 6 Common Core Standards Mathematics Grade 6 Common Core Standards Launch Search In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. 1. Students use reasoning about multiplication and division to solve ratio and rate problems about quantities.

By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates. 2. Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems. Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. 3. Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms.

Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities. 4. Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. Students recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability.

Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected. Students in Grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They prepare for work on scale drawings and constructions in Grade 7 by drawing polygons in the coordinate plane. Domain Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Note: Expectations for unit rates in this grade are limited to non-complex fractions.) 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b.

Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. The Number System Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.). How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12.

Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite. b.

Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 6.NS.7 Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3° C > –7° C to express the fact that –3° C is warmer than –7° C. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than $30. 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2. 6.EE.3 Apply the properties of operations as strategies to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply properties of operations to y + y + y to produce the equivalent expression 3y. 6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities. 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Geometry Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.1 Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism.

Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Statistics and Probability Develop understanding of statistical variability. 6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages. 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values using a single number, while a measure of variation describes how its values vary using a single number. Summarize and describe distributions. 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Question: In mathematics what term is given to the middle value in a series of values ordered by size?
Answer: Median